Method for improving a manufacturing process

ABSTRACT

A factorial experiment is conducted on a manufacturing process to generate a response matrix. The responses are used to calculate individual contrasts in a document as well as replicates effects. The contrast sums are also calculated and displayed in the document. The largest of the contrast sums are identified, and effects associated with those contrast sums are tested for significance using an end count method. The information from the process transformed into “significant effects” information is used to adjust process variables to improve the manufacturing process by avoiding the effect or imparting it to a measurable response of the process.

RELATED APPLICATION(S)

This application is a continuation of U.S. patent application Ser. No.11/624,574, filed Jan. 18, 2007, which is a continuation of US patentapplication Ser. No. 10/842,939, filed May 20, 2004, now abandoned,which is a continuation of U.S. patent application Ser. No. 10/775,313,filed Jan. 31, 2001, now U.S. Pat. No. 6,748,279, all of which areincorporated herein by reference in their entireties.

BACKGROUND OF THE INVENTION

In the improvement of manufacturing processes and products it is oftennecessary to employ empirical methods or techniques. In most basicterms, this typically involves observing the effects of variables in aproduct or process and using the information observed from those effectsto adjust or manipulate the variables, resulting in an improved orsatisfactory product or process. However, where there are many variableswith a multitude of possible effects on the process or product, arrivingat improvements is more difficult.

Industrial methods of design and analysis of experiments have beendeveloped to assist in transforming data and improving manufacturingprocesses. However, in practical applications, field experience hasshown that existing methods do not yield adequate solutions. There is aneed for a simple and easy to use method that transforms experimentalfield data into more revealing and practical information that can beused to improve processes and products.

SUMMARY OF THE INVENTION

The present invention provides a method of manufacturing or improving amanufacturing process. In addition, the method can be applied in thedesign of a manufacturing process or product.

In one embodiment described herein, a full factorial experiment isconducted with a plurality of process variables with each of thevariables being tested at a plurality of settings, in a plurality ofcombinations of settings. Measurements of the response of the processfor each combination of level settings are recorded.

The responses of the full factorial experiment are used to calculateindividual contrasts for each process variable and each interactionamong the process variables. The individual contrasts are each displayedat a particular location in a document, or other form of display,corresponding to a particular notation. The notations indicate the levelsettings of the other of the process variables not involved in theparticular contrasts.

The individual contrasts of each process variable and each interactionare added to generate separate contrast sums which are also displayed inthe document. In addition, effects estimates for each of the contrastsums are displayed.

Contrast sums are identified that are greater than at least one of theother contrast sums by a factor of about 2. If the contrast sum is thatof an interaction effect between a plurality of process variables, theinteraction is verified by referring to the document. The documentprovides information as to whether both variables of the interactionmust be set at the levels of the interaction to impart an effectsubstantially equal to the effect of the interaction.

Furthermore, when at least two trials for the full factorial experimentare conducted, replicate effects can be generated. The document can beused to generate replicate effects wherein at least one hypotheticaladditional process variable is assumed and one set of the trailresponses are substituted as responses for the hypothetical variable atone of two levels. Individual contrasts for the hypothetical variableare calculated, including the interaction contrasts thereof, to generatereplicate effects.

Contrast sums are identified that are both greater than the next largestcontrast sum by a factor of 2, as well as greater than all replicateeffects calculated. Of the identified contrast sums, the significance ofthe contrasts, or associated effects, are tested using an end countmethod. Higher order effects are tested first.

In order to test the higher order effects, the lower order effects aretemporarily removed. If an effect is found to be significant, it ispermanently removed before testing the significance of remaining effectsassociated with identified contrast sums.

The raw information from the process is thus transformed intoinformation regarding the “significant effects” of level settings of theprocess variables. The level settings of the process can be adjusted toimpart the “significant effects” to the process, or to avoid them,depending on whether the effects shift the process in the direction ofan improvement.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow sheet showing the steps of an embodiment of the method.

FIG. 2 is the response matrix for Example #1.

FIG. 3 is the worksheet used to calculate and display individualcontrasts as well as contrast sums.

FIG. 4 is the worksheet of FIG. 3 completed for Example #1.

FIG. 5 is a Pareto chart of the contrast sums calculated in FIG. 4.

FIG. 6 is a graph of the responses of cells (1), a, c, and ac of theresponse matrix of FIG. 2, for Example #1.

FIG. 7 is the graph of FIG. 6 with the AC interaction effect removedfrom the ac response.

FIG. 8 is the worksheet of FIG. 4 recalculated after the AC interactionhas been removed for Example #1.

FIG. 9 is a Pareto chart showing the contrast sums of FIG. 8.

FIG. 10 is a response matrix for Example #2.

FIG. 11 is a the worksheet of FIG. 3 completed for Example #2.

FIG. 12 shows how the variables in the Yates method table of FIG. 13 arecalculated for Example #2.

FIG. 13 is a table showing the results of the Yates method for Example#2.

FIG. 14 is a block diagram of a general purpose computer for use withthe method.

FIG. 15 is a representation of the “plane” discussed in Example #3.

FIG. 16 is a graph of the responses discussed in Example #3.

FIG. 17 is FIG. 16 with the AB interaction removed from the ab cell.

DETAILED DESCRIPTION OF THE INVENTION

The present invention relates to a method of manufacturing or improvinga manufacturing or fabrication process, or a product or article. Thevarious embodiments of the method provide a way of transforming rawinformation regarding key variables and the impacts thereof on theproduct/process, into focused estimates of “significant effects” thatthe input variables have on the key parameters of the process/product.Once the transformation of information takes place, the new informationis used to adjust the input variables, resulting in an improved orsatisfactory process or product.

As illustrated in FIG. 1, one embodiment of the method comprises thefollowing steps: 1) determine the input variables of the process thatmay effect the process or product parameters of interest; 2) design theexperiment and determine the passing end count required; 3) set levelsof the variables in the process according to the design of theexperiment and measure the process or product parameter; 4) calculatethe estimated effects as individual contrasts and display the effects ina worksheet; 5) determine which group of effects to test forsignificance as well as the order in which the effects will be tested;6) if the effect to be tested for significance is an interaction,temporarily remove the estimates of any lower order effects from theresponses; 7) test the effect for significance; 8) permanently removethe estimated effect if significant; 9) if the effect removed is aninteraction then recalculate the worksheet; 10) determine if the largestremaining contrast sum should be tested for significance; 11) iteratesteps 6 through 10 above; 12) use the information transformed to adjustthe input variables to impart an improvement in the process/product.

The first embodiment of the method is best illustrated by describing itin conjunction with a simplified example application. This is done inExample #1 below.

Example #1

The following first example description is directed toward improving amanufacturing or fabrication process, specifically, improving quality ofan article made by the manufacturing process. Improving product qualitymay typically entail meeting product specifications, exceeding productspecifications, or increasing the amount or percent of units of productthat meet specifications. The steps of the method recited above aredescribed in detail below and applied to the example.

For Step 1, it is determined that there are 3 manufacturing processinput variables that are likely to have effects on product quality. Theproduct quality is measured by an output response, or a productcharacteristic, with the measurement being a gage of the product qualityimprovement sought. It may be desired to target a range of values forthe product characteristic, or a single value. The productcharacteristic measured could be, for example, a measured tensilestrength of the product or component of the product. Again, the productcharacteristic can be any parameter identified as important to theproduct. The input variables, or process variables, are physical oroperating conditions of the manufacturing process, process steps, orspecifications of parts /materials used in the process such as equipmentor raw materials.

In this simplified example, each of the process variables will be testedat only 2 levels, conditions, or settings. For example, if one of theprocess variables is a temperature parameter, it may be tested at twotemperatures, or if it is, for example, a specification on a part usedin the process, it may be tested at both extremes of the currentspecification limit.

Step 2 is to design the experiment and determine the passing end count.The experimental design applied in this illustration is a traditionalfull factorial. Full factorial experiments, with P number of factors, orinput variables, each tested at X number of levels, or settings, willrequire X^(P) number of measurements of the output response to completeone full factorial experiment. In this example, there are P=3 processinput variables to be tested at X=2 levels, or settings, each. Thus, theoutput response must be measured 2³=8 times per experiment, to completethe full factorial experiment, which results in every combination offactor and level settings being tested once. To acquire the relevantdata, on line (operational) changes are made to the process variables ofinterest during manufacturing. The intent of making the changes is toestimate the impact of the variables on the output response, or productcharacteristic, and to then make adjustments to the process variables toimprove the response, or product quality based on informationtransformed into “significant effects” information by the method. Datais limited as it is desired to minimize disturbances to themanufacturing process, so that a minimal number of changes can be madeto the variables for testing purposes. The data is thus generatedaccording to the pre-designed full factorial experiment structurediscussed above to maximize the information yielded by the data. Theexperiment in Example #1 is run twice to gather 16 output responses asto product characteristic. Thus there will be a first and second set ofoutput responses, or repeat tests or trials, for each combination oflevel settings.

In accord with traditional notation used with analysis of full factorialexperiments to help simplify tracking and recordation of experimentalresults, each of the level settings for each process variable isrepresented by − or +. In addition, the process variables themselves arerepresented by A, B, or C. For example, A+ corresponds to the first ofthree process variables, set at the + level.

FIG. 2 is a response matrix and illustrates how the full factorialdesign of the experiment in Example #1 can be illustrated in matrix formusing the notion described above. The response matrix of FIG. 2 is for a2³ full factorial experiment for Example #1. The matrix is configured toreflect the design of the experiment and provide a convenient way torecord the output responses (product characteristics). The cells areeach labeled in a lower right hand corner ((1), a, b, ab, c, ac, bc, andabc) in accordance with traditional or standard cell notation forordering combinations, used with factorial experiments. Each cellrepresents a particular and unique combination of level settings for theprocess variables in the experiment. This can be seen directly from thestructure of the table, and is reflected in the notation for the cell.For instance, the ab cell is positioned in the A+ column, the B+ row,and the C− half of the response matrix. The ab notation indicates thatthe A and B variables are set at the + level.

The passing end count must also be determined in Step 2. For Example #1,a confidence level of 95% is chosen and this will later be tested by theend count. The end count is a way to verify the statistical significanceof the effects calculated from the experimental data. The mechanics ofchecking end count are discussed in more detail in Step 8.

Step 3 requires changing the process variables in accord with the designof the experiment. During the experimentation in Example #1, the processvariables, or input variables, are each set according to the design ofexperiment reflected in FIG. 2. For the first cell in the upper lefthand corner of the response matrix of FIG. 2, labeled “(1)”, all of theprocess variables are set to the − level since (1) does not correspondto any of the letters of the input variables. The tester measures theresulting product characteristic, and records the result in cell (1).This process is repeated for each of the cells. For example, for thelast cell, labeled “abc” in the lower left corner of the table, allthree of the process variables are set to the + level. The settings ofthe variables are represented by A+, B+, and C+. When all of the cellshave been filled with the appropriate output response, or productcharacteristic measurement, a full factorial experiment has beenconducted. Pairs, or repeat tests, or trials, are conducted for eachcombination of level settings of the process variables, and thecorresponding responses are recorded in pairs in the cells of FIG. 2.The product characteristic measured for each combination of levelsettings for the process variables for Example #1 are displayed in FIG.2.

Step 4 is to calculate individual contrasts for each of the changesbetween levels in the variables, and effects of the variables on theproduct characteristic. This can be done in the form of the worksheetshown in FIG. 3.

The three leftmost columns of the worksheet are labeled “2-Factors,“3-Factors,” and “4-Factors.” Each of the cells in those columns arelabeled to correspond to cells of a related response matrix. In the3-Factor column, the cells are labeled with standard notation torepresent the cells of a 3-factor response matrix, such as in Example#1. The fourth column from the left in the worksheet, labeled “Y”, isfor recording the output response of the process, in this case, themeasurement of product characteristic. For Example #1, the productcharacteristic measurements for each cell of the response matrix of FIG.1 are recorded in the “Y” column in the order indicated by the cellnotation under the “3-Factors” column.

The remaining cells of the worksheet display contrasts. The contrastsare estimates of the effects of changes in the level settings offactors, or process variables A, B, and C in Example #1. The contrastshave an equal number of + and − signs and are combinations of theresponses, or product characteristics. Each of the columns displayscontrasts for a particular factor or combinations of factors, asindicated at the top of each column by the factors, or process variablesshown. For example, the first column is labeled the “A” column toindicate that the column only displays single factor contrasts forvariable A. Single factor contrasts are displayed for each factor in theworksheet, and estimate an effect of a change in the level of the factorwith the other factors are set at either the − or + level during thechange. Two factor interaction contrasts are also displayed thatestimate the effect of changes of a factor on the effect of changes ofanother factor. Three factor interaction contrasts are also displayedthat provide estimates of the effect of changes of a factor on a twofactor interaction.

To better illustrate the physical meaning of contrasts, note that thecontrast in cell B1, in the upper left corner of FIG. 3, is representedby the notation b−(1), as indicated in the cell. This is equivalent tothe difference between the output response (product characteristic) withB set to the +, and the output response with B set to the − level, whilethe other factors are set at the − level. In addition, cell B2 in theworksheet, positioned just below cell B1, is represented by the notationab−a, which is equivalent to the difference between the output responsewith B set at the +, and the output response with B set at the − level,with A at the + level and C at the − level. To illustrate the physicalmeaning of an interaction, or contrast involving two factors, note thatcell AB 1 of the worksheet of FIG. 2 provides an estimate of the effectof a change in the level of A, on the estimated effects of changes inthe level of B discussed above. Hence, cell AB1 is represented by B2−B1which is equivalent to the difference between cell B2 and cell B1 of theworksheet. Each of the cells of the worksheet are calculated in thismanner according to the notations in the worksheet cells. Contrasts aredisplayed for each single factor change, as well as for eachinteraction, including higher order interactions involving 3 factors.

The worksheet in FIG. 3 is directed toward an experiment with 16 totaloutput response data points and only 2 to 4 factorial experiments.However, the worksheet can be expanded as needed.

The four rows at the bottom of the worksheet display: 1) the sum ofcontrasts for cells in that column (Contrast Sum); 2) the orthogonalestimate, or contrast sum divided by half the number of outputresponses; 3) the number of individual effects, or contrasts, in thecolumn (# of Estimates); and 4) the “effect estimate,” which is theaverage estimated effect, or contrast for the column.

FIG. 4 shows the worksheet completed for Example #1, using the measuredproduct characteristics from the response matrix in FIG. 2. Note thatsince there are only 3 process variables, the columns for contrastsinvolving changes in a D variable do not have physical meaning exceptfor measuring “noise” or variation not associated with the effects beingestimated. The contrasts calculated in those columns are calledreplicate effects. D is treated as a hypothetical process variable, andthe “noise” contrasts, or replicates effects, involving changes in thelevel of D are calculated by substituting the second of the repeat setof output responses, which begins with 10 under the “Y” column of theworksheet, for the hypothetical responses that would be generated by theD variable at the +level. This is illustrated for Example #1 by the D1cell of the Worksheet, which is notated in FIG. 3 as d−(1). That cell iscalculated as (1)−(1), wherein pairs of responses (productcharacteristic measurement), recorded in cell (1) of the response matrixfor the repeat tests, are subtracted from one another to reveal ameasurement of variation not attributable to the effects being tests.

For Example #1, as can be seen in FIGS. 3 and 4, there is a first set,or column, of intra-cell replicate effects under column “D,” thatmeasures variation between the repeat tests, or the variation betweenoutput responses within the cells of the response matrix of FIG. 1. Inaddition, there are second set replicate effect, columns “AD,” “BD,” and“CD,” that measure variation between the intra-cell replicate effects,the set being represented by interactions between the hypothetical Dprocess variable and each of the A, B, and C process variables. Lastly,there are third sets of replicate effects that measure variation betweenthe inter-cell replicate effects, represented by hypotheticalinteractions between D and two of the other three variables. For Example#1, FIG. 4 shows all of the cells in the D columns calculated. All ofthose numbers represent variation not attributable to the effects beingtested and are replicate effects. Again, if a process variable D wasbeing tested, these cells would be individual contrasts and notreplicate effects.

The variation in contrast sums as well as in contrasts is inspected inthe worksheet. A large variation in contrasts within a particular columnof the worksheet can be an indicator of an interaction. In FIG. 4 forExample #1, under column A, the range of estimated effects, orcontrasts, for level changes in the process variable A vary from between−8 to 16. This is indicative of an interaction between A and anothervariable. By cross referencing, or comparing, the contrasts in FIG. 4 tothe notations in the worksheet of FIG. 3, it can be seen that theinteractions are between the A and B process variables. For example, incells A1 and A2, in FIG. 4, the estimated effect on the productcharacteristic is −8 and −7, whereas for cells A3 and A4, the estimatedeffects are 16 and 16. In FIG. 4 it can be seen from the contrastnotations for cells A1 and A2, a−(1) and ab−b, that the C variable isset at the − level for both contrasts. However, for cells A3 and A4, thecontrast notations indicate that the C process variable is set at the +level. This is indicative that C should be set at the + level while thesetting of A is forced from the minus to the plus level.

Step 5 is to identify or determine which estimated effects of theprocess variables should be tested for significance. The sum ofcontrasts displayed for Example #1 in FIG. 4 are plotted on a ParetoChart in FIG. 5 to assist in deciding which estimated effects to test,and which to exclude. Adjacent contrast sums that drop by about a factorof two or more on the Pareto Chart are noted. Contrast sums on the highside of the drop are identified and tested for significance. Those onthe low side of the drop may be excluded. Furthermore, for 2 and 3factor experiments, the contrast sums of the replicate effects arecompared with the contrast sums determined to be on the high side of thedrop. If the replicate effects are approximately equal to the contrastsums identified, those identified contrast sums may be excluded fromtesting, since the background noise would appear to be as large as theeffects.

As seen in FIG. 4, for Example #1, the first drop off, or break, ofabout a factor of 2, is between the A contrast sum and the BC contrastsum. Thus the AC, C, and A contrasts will be tested for significance. Inaddition, the highest order contrasts are tested first. AC is thehighest order in Example #1 and should thus be tested for significancefirst.

Step 6 requires that before an interaction effect is tested forsignificance, the estimates of all lower order effects involved in theinteraction are temporarily removed from the response matrix to isolatethe effect of the interaction. For Example #1, the effects of processvariables A and C must be removed to test for the significance of thecontrasts for the AC interaction. When removing lower order effects,such as those of process variables A and C, the orthogonal estimates,calculated and displayed at the bottom of the worksheet in FIG. 4, areused. The orthogonal effects are removed by subtracting the orthogonalestimate from product characteristics (output responses) that correspondto the + level settings for the process variables A and B. This isillustrated in Table 1 below. The Y column of Table 1 displays theproduct characteristic measurements that correspond to the indicatedcell of the response matrix of FIG. 2. The orthogonal estimates forprocess variables A and C are 4.375 and 6.375, as shown in FIG. 4 forExample #1. These are subtracted from the product characteristicmeasurements as shown in Table 1, and as explained above, to arrive atthe results in the last column, which shows the product characteristicswith the orthogonal estimates removed.

TABLE 1 SUBTRACTING ORTHOGONAL ESTIMATES FOR A AND B FROM THE PRODUCTCHARACTERISTICS Y. Orth. Est. Orth. Est. Y with Orthogonal Cell Y A forA C for B Estimates Removed (1) 12 − − 12 a 4 + 4.38 − −0.38 b 12 − − 12ab 5 + 4.38 − 0.62 c 6 − + 6.38 −0.38 ac 22 + 4.38 + 6.38 11.24 bc 6 − +6.38 −0.38 abc 22 + 4.38 + 6.38 11.24 (1) 10 − − 10 a 3 + 4.38 − −1.38 b11 − − 11 ab 4 + 4.38 − −0.38 c 7 − + 6.38 0.62 ac 23 + 4.38 + 6.3812.24 bc 5 − + 6.38 −1.38 abc 21 + 4.38 + 6.38 10.24

Step 7 is to test the estimated effect for significance, in this case,the interaction effect. The method used is an end count. To do this, theresponses, or product characteristic measurements, are sorted in rankorder (ascending order) and all associated cells in the Table 1 that arein the same row as the sorted response cell, are also shifted with theassociated response cell. This is shown in Table 2 below. Table 2 hasone more column than Table 1. The additional column is the rightmostcolumn in the Table 2 and displays the product of the level settings forprocess variables A and C. AC is thus only positive when either bothprocess variables A and C are positive, or both are negative. Thesignificance of this is that it is indicative of whether the levels ofthe variables are set to permit an interaction. The separation between +and − signs in the AC column in Table 2 is indicative of the amount ofoverlap between the responses with potential AC interaction and thosewithout potential AC interaction. As such, an end count is used toquickly gage the significance of the AC interaction. The end count isdone by first counting − signs from the top of the AC column until a +sign is encountered. Next, + signs are counted starting from the bottomof the column until a − sign is encountered. The two counts are addedtogether to get an end count. Table 2 shows that the end count for ACfor Example #1 is 16. Table 3 shows that an end count of 10 is requiredfor a confidence level of 95%. The AC interaction is thus identified assignificant.

TABLE 2 TABLE 1 SORTED IN RESPONSE RANK ORDER Y with Orthogonal Orth.Est. Orth. Est. Estimates Cell Y A for A C for C Removed AC a 3 + 4.38 −−1.38 − bc 5 − + 6.38 −1.38 − a 4 + 4.38 − −0.38 − c 6 − + 6.38 −0.38 −bc 6 − + 6.38 −0.38 − ab 4 + 4.38 − −0.38 − ab 5 + 4.38 − 0.62 − c 7 − +6.38 0.62 − (1) 10 − − 10 + abc 21 + 4.38 + 6.38 10.24 + b 11 − − 11 +ac 22 + 4.38 + 6.38 11.24 + abc 22 + 4.38 + 6.38 11.24 + (1) 12 − − 12 +b 12 − − 12 + ac 23 + 4.38 + 6.38 12.24 + EC = 16

TABLE 3 REQUIRED ENDCOUNT Required Endcount given the confidence listedbelow: # Factors 90% 95% 99% 99.9% 2 8 9 11 14 3 9 10 12 16 4 10 11 1316

Step 8 is to permanently removed the estimated effect if significant.The estimated effect of the interaction of AC must be removed to testfor significance of the remaining identified effects, process variablesA and C. The original product characteristic measurements are used forthis, from the response matrix in FIG. 2, that is, the lower ordereffects that were removed earlier must be replaced. The estimatedeffects of −AC can be mathematically removed by directly subtracting oradding it to any cells in the ac matrix. However, the estimate should beremoved to achieve the smallest remaining estimates for the lower ordereffects. For Example #1, a graph is created to aid in removing theestimate of the AC interaction to achieve the smallest remainingestimates. This graph is illustrated in FIG. 6. The graph indicates thatremoving the AC interaction effect from cell ac will leave the smallestA and C effects.

FIG. 6 shows that there is not a perfect spike interaction between the Aand C process variables in Example #1. A perfect spike interaction wouldhave an estimated effect close to zero at one of the levels of A and alarge estimate at the other level of A. In this case, the C effect is −5at the A− level and 20 at the A+ level.

FIG. 7 is the graph of FIG. 6 with the estimated effect for AC removedfrom the ac response. By doing so, the unequal sensitivity of the Afactor when C is set at the + level rather than the − level, has beenset to one of two levels. Now when the effect of A is estimated, it isestimated when C is set at the − level. Also, when the effect of C isestimated, it is done with A set to the − level.

Factors involved in a removed interaction are set to either the + or −level. Examples of the possible settings are summarized in Table 4below.

TABLE 4 MAIN EFFECT SETTINGS IN THE MATRIX AFTER AN INTERACTION HAS BEENREMOVED Remove AC average A effect is estimated C effect is estimatedeffect estimate from: with C set to: with A set to: (1) C+ A+ a C+ A− cC− A+ ac C− A−

TABLE 5 RESPONSE TABLE FOR Y WITH THE ESTIMATE OF AC REMOVED FROM CELLAC (AND ABC) Y without Original Estimate of the estimate Cell Y AC of AC−1 12 12 a 4 4 b 12 12 ab 5 5 c 6 6 ac 22 23.25 −1.25 bc 6 6 abc 2223.25 −1.25 −1 10 10 a 3 3 b 11 11 ab 4 4 c 7 7 ac 23 23.25 −0.25 bc 5 5abc 21 23.25 −2.25

For Example #1, the AC interaction effect is now removed from theresponses using the average estimated effect (not the orthogonalestimate), as shown in Table 5 above.

Step 9 is to recalculate the worksheet if the effect removed is aninteraction. Because the last estimate removed was for an interactionbetween process variables A and C, the worksheet is recalculated beforeproceeding to Step 11. When the removed estimate is a main effect, theworksheet is not recalculated. FIG. 8 is the worksheet, recalculatedwith the AC estimated effects removed from the product characteristicsmeasurements of the ac and abc cells of the response matrix of FIG. 1.

Step 10 is to determine if the largest remaining contrast sum should betested for significance. The contrast sums from FIG. 8 are again plottedon a Pareto Chart as shown in FIG. 9, and again checked for an adjacentdrop between contrast sums by a factor of 2 or more, as was previouslydone before the AC interaction was removed. The contrast sums on thehigh side represent effects that should be identified and tested forsignificance. At this point it is noted that if enough leverage has beenidentified to control the product characteristic, the analysis may bediscontinued. Also, if the factor to be checked for significance is acomponent of an interaction where the effect of the interaction willdetermine the setting of the factor to be checked, then the analysis maybe discontinued, since no degrees of freedom for the variable remains.For Example #1, analysis may be discontinued since the remainingvariables that appear on the high side of the break on the Pareto Chartin FIG. 9 are variables in the interaction AC. Assuming enough leveragehas been identified with the interaction, then the results of thisexperiment may cause A and C to be both set at minus levels if it weredesired to keep the process characteristic low. Nonetheless, forpurposes of illustration, the analysis will continue.

Table 5 is reorganized in rank order response, shown in Table 6. This isdone in the same manner as was previously done when the lower ordereffects of the A and C variable were removed, in Table 2.

TABLE 6 TABLE 5 IN RANK ORDER, SHOWING A SETTING LEVELS Y w/o theEstimate Cell of AC A Level abc −2.25 + ac −1.25 + abc −1.25 + ac−0.25 + a 3 + a 4 + ab 4 + ab 5 + bc 5 − c 6 − bc 6 − c 7 − −1 10 − b 11− −1 12 − b 12 − EC = 16

The end count is taken using Table 6. The end count is 16 since there isno overlap between the + and − signs of the A level column. This exceedsa required endcount of 10. A is thus found to be significant with 95%confidence.

The A process variable effect is then permanently removed by subtractingthe orthogonal estimates from the responses in FIG. 7. This is shown inTable 7 below.

Step 11 is to begin again at step 6. However, the worksheet does notneed to be recalculated at this stage because the effect of the Aprocess variable is a main effect and has been removed from the arrayorthogonally. This means that C effect is still the third largestcontrast sum (−42) and should be the next one checked for significance.The endcount check for C is shown in Table 7 and Table 8 below.

TABLE 7 REMOVE THE EFFECT OF A TO CHECK END COUNT FOR C: Y w/o AC effector Cell Y w/o AC effect A Effect of A A effec

−1 12 − 12 a 4 + −7.25 11.25 b 12 − 12 ab 5 + −7.25 12.25 c 6 − 6 ac−1.25 + −7.25 6 bc 6 − 6 abc −1.25 + −7.25 6 −1 10 − 10 A 3 + −7.2510.25 B 11 − 11 Ab 4 + −7.25 11.25 C 7 − 7 Ac −0.25 + −7.25 7 Bc 5 − 5Abc −2.25 + −7.25 5

indicates data missing or illegible when filed

TABLE 8 TABLE 7 SORTED IN RANK ORDER WITH THE C LEVEL ADDED: Cell Y w/oAC and A Effects C Level bc 5 + abc 5 + c 6 + ac 6 + bc 6 + abc 6 + c7 + ac 7 + −1 10 − a 10.25 − b 11 − a 11.25 − ab 11.25 − −1 12 − b 12 −ab 12.25 − EC = 16

The endcount of 16 exceeds the required endcount of 10. C has been foundto be significant with 95% confidence.

The new and transformed information yielded is that the largest effectis AC with an estimated effect of 23.25. Setting both process variablesA and C to the + levels causes an increase of about 23 in the productcharacteristic. Furthermore, when process variable C is set to the −level, the A effect is significant with an estimated effect of −7.25.Also, when the process variable A is set to the minus level the C effectis significant with an estimated effect of −5.25. Thus, in order tomaximize the product characteristic, or output response, both A and Cmust be set to their plus levels. To minimize the productcharacteristic, either and or both A and C should be set at the minuslevel.

Step 12 is to use the information that has been transformed from processdata into information that can be used to directly control the process,to improve the product/article of manufacture, by setting the variablesas a function of the “significant effects.” It should be determinedwhether any of the significant effects, estimated by the contrasts, willimpart a shift in the product characteristic in the direction desired,or whether the effect is to be avoided. Also, it is noted that thedesired product characteristic may be a range of values. If theestimated effects are indicative of level settings of the processvariables that will improve the process as whole, taking intoconsideration costs and other factors associated with maintaining thelevel settings, then the factors may be set at the appropriate levelsettings to impart the estimated effects. For Example #1, if the productcharacteristic is, for example, percent impurity of some component, andit is desired to derive a more pure product, both A and C will be set atminus levels if not cost prohibitive. In that way, even if one variablegoes out of control, the other variable may remain at the minus level,preventing the interaction effect from occurring between the variables,causing a high level of impurity. On the other hand, if the productcharacteristic is, for example, tensile strength, and it is desired tohave a strong product with high tensile strength, both A and C may beset at their plus levels if it is not cost prohibitive.

Example #2

Example #2 compares an embodiment of the method to the Yates analysis.Example #2 is also directed toward improvement of a fabrication processwhere a spike interaction is present between variables. Example #2, likeExample #1, is an alternative embodiment of the method and is alsomerely one example application of the method.

In Example #2, for Step 1, two input variables are selected for testingat 2 levels each. Again, a product characteristic is the measuredresponse or output.

Step 2 is to design the experiment using a factorial design. In Example#2, there are 2 factors in the experiment with 2 levels each. A 2²response matrix is thus required. Each combination of level settings forthe variables is to be tested four times, to produce four repeatresponses in each cell of the response matrix.

A passing end count is determined in accordance with Step 3 depending onthe confidence required.

Step 4 is to set the levels of the variables and record the responses tocomplete the full factorial experiment with four repeat runs. Theresults of the experiment are shown in FIG. 10.

Example #2 is a simplified example and FIG. 10 of the example shows thatan interaction is occurring between process variables A and B, since theresponses in the ab cell are larger than the responses in the othercells.

Step 5 is to calculate the estimated effects as individual contrasts anddisplay the effects in the worksheet. This is shown in FIG. 11.Evaluating the contrasts in the worksheet reveals that there is aninteraction between A and B when both variable are set at + levels.First, the set of individual contrasts shown in the worksheet for Arange from 1 to 10. The set of contrasts for B range from −1 to 7.Finally, the set of contrasts for the interaction between A and B is 6to 8. The range of contrasts in the A and B process variables combinedwith steady AB contrasts of limited variance is one indication of an ABinteraction effect. This can be verified by examining the leftmostcolumn of the worksheet in FIG. 11 for the particular row in which eachcontrast is displayed, which provides a notation for the row thatindicates the level setting of variables not involved in the particularcontrast. This is best seen in the worksheet for Example #2 in FIG. 11,showing that the value of each contrast for process variable A is at thehigh end of the range in each “ab” row, the “ab” notation indicatingthat B is set at the + level. Furthermore, each value for the contrastsfor process variable A is at a low end of the range in each “a” row, the“a” notation indicating that B is at the− level. The AB interaction canbe verified by examining the B level settings in the same manner.

An interaction can thus be predicted based only on the worksheet, andthe level settings of process variables A and B may be set to impart theAB interaction effect to the response, or to avoid it, depending on thetarget value of the response.

An application of the well known Yates analysis to Example #2 is shownin FIGS. 12 and 13. The column labeled “Effects” in these figures showthe calculated effects for A, B, and the AB interaction, in that orderfrom the top of the column. The A effect and B effect are calculated tobe 5 and 4, with the AB effect calculated to be only 3.5. As can be seenin this simple example in FIG. 9, it is clear that the response for ABis the largest with all other responses approximately equivalent. It isthus clear that the Yates method is yielding the wrong result and the Aeffect is not the largest effect.

Example #3

Example #3 provides further explanation of an embodiment of the methodas applied to a spike interaction. Example #3 is again directed towardimprovement of a manufacturing process, having process variable A, andB, with two level settings, and a measurable response indicative ofimprovement to the process.

As has been shown in the description of the embodiment of the method inExample #2, a method is provided to analyze full factorial experimentsto identify and quantify spike interactions. Spike interactions can beexplained by viewing a 2̂2 experimental matrix plotted as a plane.

To explain a spike interaction it is helpful to picture a plane createdin space having 4 corners, as illustrated in FIG. 15. For Example #3,the x, y component of each corner are determined by the settings ofprocess variables A, B. The z component of each corner is set by themeasurable response, which is equivalent to the height of each corner ofthe plane.

If the responses of all cells are approximately equal and are, forexample, 2 units, the plane will float 2 units above the zero plane andwill be parallel to the zero plane. For Example #3, there is an A maineffect of 0 units, so corners (1) and a will be the same, in this case 4units off the zero plane. There is also a B main effect of 2 units, socorner b will be 2 units higher than corner (1). If there is nointeraction corner ab will be equal to corner (1) plus both the A and Beffects. In this case that would yield a corner ab at 6 (4+0+2). Ifthere is no interaction the main effects are superimposed upon eachother, and the plane remains flat, but no longer parallel to the zeroplane. However, for Example #3, there is a spike interaction. This isshown in FIG. 16, wherein the response of the ab cell is 22. Thus, A andB are interacting, at one level, to display a higher response thansimply superimposing the A and B effects. The responses are (1)=4, a =4,b=6, and ab=22.

Interactions impart a twist on the plane. Traditional interactions causeopposite cells to move as a pair. For example a traditional ABinteraction will cause cells (1) and ab to both move in the samedirection. Traditional interactions cause the plane to look like asaddle. Main effects superimposed over traditional interactions willcause the plane to look like a tilted saddle. The Yates analysis isbased on the analysis of traditional interactions.

Field experience has proven the existence of spike interactions. Spikeinteractions do not effect the response plane in the same manner astraditional interactions. Spike interactions cause one cell of thematrix (not two) to move independent of the other cells. For example, apositive ab spike interaction will cause the ab corner of the plane to“spike up” making it significantly higher than the other 3 corners ofthe matrix. The (1) corner which is traditionally paired with the abcorner is unmoved by the effect of the spike.

A perfect spike interaction yields contrast sums for both theinteraction and the two associated main effects which are equal withinmeasurement error. For example, a perfect AB spike will result incontrast sums of AB, B, and A all being approximately equal. This is whyhigher order interactions are tested first. For the purposes of theembodiment of the method in Example #3, spike interactions include bothperfect spike interactions and approximated spike interactions.

Recognizing a spike interaction is one reason why, in step 8 of theembodiment of the method shown in Example #1, the AC interaction effectwas removed from only one response, the ac response. While the effect ofthe interaction can be mathematically subtracted from both cells (1) andac using the orthogonal estimate instead of the effect estimate thisdoes not accurately represent what is physically happening. When a spikeinteraction is subtracted from more cells than is physically warrantedthe remaining contrasts are artificially large.

For the present Example #3, FIG. 16 should be used to remove the ABinteraction when permanently removing its effect. The AB effect is 16,and removing it from the ab cell will achieve the smallest remainingeffects, which is in fact, the location of the where the spikeinteraction occurs. FIG. 17 shows Example #3 with the AB effect removedfrom the ab cell.

By removing the AB effect from the ab cell, the effect estimate of A isnow made with B set to the minus level, and the effect estimate of B isnow made with A set to the minus level. This yields useful informationin that, since AB has been shown to be the interaction of interest, itwill be most desirable to also know the effect of either variable alonewith the other set so as not to interact in the spike interaction. Thus,by graphically representing the responses, and removing the interactioneffect to achieve the smallest remaining effects, useful information isobtained that can be directly used to determine settings for processvariables. The same considerations may be given to where to set thevariables as was discussed in Step 12 of Example #1.

Example #4

A full factorial experiment was run for a manufacturing process whereinelectrical components were being manufactured. Finished parts werefailing dielectric testing. Three variables were identified as possiblecontributors to the problem. The variables were tested using athree-factor full factorial experiment. The response was, arc-volts, thevoltage at which the part failed.

The present method identified an AB spike interaction when variables Aand B were set at a low value (−). The spike interaction provided theneeded response level. Given the consequences of building a weak part,and the cost of setting both A and B to the low level, it was decided toset both A and B to the low level.

Examples #1, #2, #3, and #4, have been directed toward the improvementof a manufacturing process to yield improved product characteristic.Manufacturing processes can include but are certainly not limited to,manufacturing of vehicles parts, vehicles, general electronic apparatusand devices, computers, computer components, scientific apparatus,medical apparatus, chemicals, machinery, foods, construction materials,tools, pharmaceuticals, paper goods and printed matter, paint, rubbergoods, leather goods, furniture, housewares, cordage and fibers,fabrics, clothing, fancy goods, toys and sporting goods, and beverages,cosmetics and cleaning preparations, lubricants and fuels/oil, generalmetal goods, jewelry, firearms, musical instruments, and even theprocessing of natural goods. However, as will be appreciated, theembodiments of the method have broad applicability. The output responsesmonitored can be any form of product or article characteristic as wellas a characteristic of the fabrication or manufacturing process itself.Thus the improvement sought and achieved through application of any ofthe various embodiments of the method can include improvements not onlyto the product or article, but also to the manufacturing or fabricationprocess. Examples of measurable responses monitored to gage improvementsto the process include production rate of the process and any efficiencyin the process.

In addition, embodiments of the method can also be used in the operationof a manufacturing process, such as, for example, when a process hastemporarily deviated from a target value required for an operatingparameter of the process, and it is desired to return the process tonormal operation. The previous settings of variables may be unknown, andhence, an embodiment of the present method can be used to return thevariables to the previous settings to attain the range sought for theoperating parameter. The operating parameter may be related to, but arenot limited to, production rates, manufacturing efficiency parameters,and product characteristics of the products generated by the process.

It will also be appreciated that embodiments of the method can beapplied to the design of processes and products. Such applications ofembodiments of the method may typically be in connection with benchscale models of a manufacturing or fabrication process or prototypes ofa product or article. Experimentation can be done on the bench scale, oron the prototypes, and an embodiment of the method can be used to selectthe correct level settings for the variables.

One skilled in the art will also recognize that the present inventionmay be implemented through the use of a general purpose computer system.For example, the contents of the worksheet of FIG. 3 may be calculatedand stored in the computer in a variety of forms including aspreadsheet, or the iterative steps of the method as well as thegraphical interpretations may be done with the computer. An embodimentof the method may be implemented by a computer system, includingreceiving and adjusting variables through the input/output devices 4,based on the information yielded by the method. In one alternativeembodiment, an embodiment of the method is implemented in the computerand a signal is sent to a controller to adjust the level settings of theprocess variables based on information transformed by the embodiment ofthe method. Any one of the embodiments of the method may also be storedon a computer readable medium, such as a memory, which can then be usedwith a computer to perform the method.

FIG. 14 is a block diagram of a general purpose computer for practicingpreferred embodiments of the present invention. The computer system 1contains a central processing unit (CPU) 2, a display screen 3,input/output devices 4, and a computer memory (memory) 5.

As the embodiments of the method can be implemented through the use of ageneral purpose computer system, wherein the particular documentsdescribed previously are not necessary, so can the documents be modifiedand embodied in various forms of display. For example, the worksheet ofFIG. 3 may be implemented in a variety of forms. A display could begenerated in more tabular form with the fields of the tablecorresponding to similar notation, or perhaps in graphical form.

From the foregoing it will be appreciated that, although specificembodiments of the invention have been described herein for purposes ofillustration, various modifications may be made without deviating fromthe spirit and scope of the invention. Accordingly, the invention is notlimited except as by the appended claims.

1. A method for improving a manufacturing process wherein there are aplurality of process variables and a value of a measurable response ofthe manufacturing process is indicative of an improvement to theprocess, the method comprising: conducting a full factorial experimentby setting a plurality of the process variables at a plurality ofsettings in a plurality of combinations of settings and receiving atleast one measurement of the response of the process for eachcombination of level settings; calculating individual contrasts for eachprocess variable and each interaction among the process variables usingthe received responses of the full factorial experiment and displayingthe individual contrasts for each variable and each interaction;verifying that both variables of an interaction contrast must be set atthe levels of the interaction to impart an effect substantially equal tothe effect of the interaction by evaluating the variance of thecontrasts displayed; setting the process variables as a function of theeffect of the verification; and operating the manufacturing process. 2.A computer-implemented method of improving a manufacturing processwherein a target is determined for a measurable response, the targetbeing indicative of an improvement in the process, the methodcomprising: conducting a full factorial experiment with at least twoprocess variables being adjusted between at least two level settingswith output responses being measurements of the response for which thetarget is determined; receiving the responses of the full factorialexperiment and using the responses to calculate individual contrasts foreach process variable and each interaction among the process variablesand displaying each of the contrasts in a document; adding theindividual contrasts of each process variable and each interaction togenerate separate contrast sums; selecting at least one of the contrastsums when it is greater than at least one of the other contrast sums bya predefined factor; and adjusting the level settings of the processvariables as a function of an estimated effect associated with theselected contrast sum.
 3. A computer readable medium for instructing acomputer to perform a method for improving a manufacturing process,comprising: receiving level settings and responses for a factorialexperiment; calculating individual contrasts for each process variableand each interaction among the process variables; and testing thesignificance of effects associated with the contrasts, wherein when aneffect is found to be significant and is an interaction effect, it isremoved before testing the significance of another affect, the removalbeing done to achieve the smallest remaining estimates for the lowerorder effects.